Understanding Parabolas: Vertex, Focus, and Directrix

The video tutorial explains how to find the vertex, focus, and directrix of a parabola given by the equation x^2 + 8x - 2y - 24 = 0. It demonstrates rewriting the equation in standard form by completing the square, identifying the vertex as (-4, -2), and calculating the focus and directrix. The focus is found to be 5 units above the vertex at (-4, 3), and the directrix is a horizontal line at y = -7. The tutorial concludes with graphing the parabola and visualizing its properties.

What is the first step in finding the vertex, focus, and directrix of a parabola?

• Calculate the derivative.

• Find the x-intercepts.

• Convert the equation to standard form.

• Graph the equation directly.

Which part of the parabola equation indicates whether it opens up or down?

• The coefficient of x.

• The coefficient of y.

• The sign of p.

• The constant term.

How does the sign of p affect the direction in which a parabola opens?

• If p is positive, it opens left; if negative, it opens right.

• If p is positive, it opens right; if negative, it opens left.

• If p is positive, it opens up; if negative, it opens down.

• If p is positive, it opens down; if negative, it opens up.

What is the purpose of completing the square in the context of parabolas?

• To convert the equation into a perfect square trinomial.

• To simplify the equation.

• To eliminate the y variable.

• To find the x-intercepts.

What is the vertex of the parabola given by the equation x^2 + 8x - 2y - 24 = 0?

• (-4, -2)

• (4, 2)

• (-2, -4)

• (2, 4)

How do you determine the focus of a parabola from its vertex?

• Add p to the x-coordinate of the vertex.

• Add p to the y-coordinate of the vertex.

• Subtract p from the x-coordinate of the vertex.

• Subtract p from the y-coordinate of the vertex.

In the context of parabolas, what does the absolute value of p represent?

• The distance from the vertex to the y-axis.

• The distance from the focus to the directrix.

• The distance from the vertex to the focus and directrix.

• The distance from the vertex to the x-axis.

What is the significance of the vertex in a parabola?

• It is the midpoint of the parabola.

• It is the intersection with the x-axis.

• It is the highest or lowest point of the parabola.

• It is the intersection with the y-axis.

What is the equation of the directrix for the given parabola?

• y = -7

• y = 7

• x = -7

• x = 7

What is the final step in graphing a parabola after finding the vertex, focus, and directrix?

• Calculate the area under the curve.

• Sketch the parabola using the vertex, focus, and directrix.

• Draw the axis of symmetry.

• Plot the x-intercepts.